$\begin{aligned} &G(x)=(x+3)^2 \\\\ &g(x)=G'(x) \end{aligned}$ $\int_{-1}^{7} g(x)\,dx=$
$g$ is the derivative of $G$, which means $G$ is an antiderivative of $g$. Since we know the antiderivative of $g$, we can use the fundamental theorem of calculus: For every function $g$ and its antiderivative $G$, $\int_a^b g(x)\,dx=G(b)-G(a)$. $\begin{aligned} &\phantom{=}\int_{-1}^{7} g(x)\,dx \\\\ &=G({7})-G({-1}) \\\\ &=({7}+3)^2-({-1}+3)^2 \\\\ &=100-4 \\\\ &=96 \end{aligned}$ In conclusion, $\int_{-1}^{7} g(x)\,dx=96$